Mixed Connectivity of Random Graphs

Abstract

For positive integers k and $\lambda$, a graph G is $(k,\lambda )$-connected if it satisfies the following two conditions: (1) $|V(G)|\ge k+1$, and (2) for any subset $S\subseteq V(G)$ and any subset $L\subseteq E(G)$ with $\lambda |S|+|L|<k\lambda$, $G-(S\cup L)$ is connected. For positive integers k and $\ell$, a graph G with $|V(G)|\ge k+\ell +1$ is said to be $(k,\ell )$-mixed-connected if for any subset $S\subseteq V(G)$ and any subset $L\subseteq E(G)$ with $|S|\le k, |L|\le \ell$ and $|S|+|L|<k+\ell$, $G-(S\cup L)$ is connected. In this paper, we investigate the $(k,\lambda )$-connectivity and $(k,\ell )$-mixed-connectivity of random graphs, and generalize the results of Erdős and Rényi (1959), and Stepanov (1970). Furthermore, our argument can show that in the random graph process $\tilde{G} = \left( {{G_t}} \right) _0^N$, $N=\left( {\begin{array}{c}n\\ 2\end{array}}\right)$, the hitting times of minimum degree at least $k\lambda$ and of $G_t$ being $(k,\lambda )$-connected coincide with high probability, and also the hitting times of minimum degree at least $k+\ell$ and of $G_t$ being $(k,\ell )$-mixed-connected coincide with high probability. These results are analogous to the work of Bollobás and Thomassen (1986) on classic connectivity.